Sunday, 1 March 2020

electromagnetism - How can a magnetic field accelerate particles if it cannot do work?



A varying magnetic field can accelerate charge particles, but it is said that a magnetic field can't do any work so it should not be able to speed up charged particles, right? How is this apparent contradiction resolved?



Answer



A varying magnetic field generates an electric field, and an electric field can do work on a particle. This is called Faraday's law of induction:


$$\nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t}$$


The full Lorentz force equation is


$$\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$$


So for example, if the magnetic field is increasing in the $\hat{z}$ direction, such that


$$\vec{B} = b t \hat{z}$$


and


$$\frac{\partial \vec{B}}{\partial t} = b \hat{z}$$



then the electric field is determined by


$$\nabla \times \vec{E} = - b \hat{z}$$


Thus the electric field is not zero, so work can be done on a charged particle as a result of a changing magnetic field.


No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...